3. Results

To illustrate how the passage of coherent structures (LMRs and HMRs) in the outer layer affects processes in the roughness sublayer, we have first generated time–height contours of fluctuating streamwise velocity, $\tilde{u} ^{\prime }(y,t)$ , for cases HR and CR (1.1). These plots are generated by recording time series of velocity across the depth of the domain (vertical red line in figure 1 a,c) (Anderson et al. Reference Anderson, Li and Bou-Zeid2015). Thus, LES is used to attain datasets that would be more commonly associated with a micrometeorological field campaign, for example, where a series of sonic anemometers are mounted on a tower. Figure 2 shows contours of $\tilde{u} ^{\prime }(y,t)$ for cases HR (a) and CR (b and c correspond with time series at positions $\{x_{1},z_{1}\}$ and $\{x_{4},z_{4}\}$ , respectively; see also figure 1 b). Positions $\{x_{1},z_{1}\}$ (figure 2 b) and $x_{4}$ (figure 2 c) were chosen for illustration given their positions above and immediately upwind of cubes (please see Anderson et al. (Reference Anderson, Li and Bou-Zeid2015) for additional visualizations). In all cases, the $\tilde{u} ^{\prime }(y,t)$ contours exhibit a clear time lag between the passage of LMRs (HMRs) in the outer layer ( $y/{\it\delta}\gtrsim 0.5$ ) and suppression (excitation) of activity in the roughness sublayer (Mathis et al. Reference Mathis, Hutchins and Marusic2011). Consider, for example, the figure 2(c) contour in the interval $15\lesssim tU_{0}/{\it\delta}\lesssim 18$ . It is apparent that the magnitude of $\tilde{u} ^{\prime }$ fluctuations increases as the aloft HMR passes by. In an earlier study, we included contours of resolved Reynolds shear stresses (streamwise–wall-normal component: $\tilde{u} ^{\prime }\tilde{v}^{\prime }$ ) owing to turbulent sweeps ( $\tilde{u} ^{\prime }>0$ ; $\tilde{v}^{\prime }<0$ ) and ejections ( $\tilde{u} ^{\prime }<0$ ; $\tilde{v}^{\prime }>0$ ) on figure 2-style $\tilde{u} ^{\prime }$ contours (Anderson et al. Reference Anderson, Li and Bou-Zeid2015). It was comprehensively shown that $\tilde{u} ^{\prime }\tilde{v}^{\prime }$ owing to sweeps and ejections was always elevated following the passage of high- and low-momentum regions, respectively, and that the largest values occurred in the roughness sublayer. Consideration of the figure 2 contours thus provides some qualitative support for arguments that an underlying AM process takes place for the present high- $Re_{{\it\tau}}$ rough-wall channel flow cases. In Anderson et al. (Reference Anderson, Li and Bou-Zeid2015), we compared local mean streamwise velocity profiles for case CR against literature and DNS results, and reported strong agreement.

Figure 2.

Time–height contours of fluctuating streamwise velocity, $\tilde{u} ^{\prime }(y,t)/u_{{\it\tau}}$ , over $tU_{0}/{\it\delta}=30$ eddy turnovers. Panel (a) is for HR at position $\{x_{c},z_{c}\}$ in figure 1(c). Panel (b,c) are for CR at positions $\{x_{1},z_{1}\}$ and $\{x_{4},z_{4}\}$ , respectively (please see figure 1 b for reference). In (b,c), horizontal orange line denotes $h/{\it\delta}$ .

Figure 3.

Time series of velocity and AM parameters (recall § 1.1 decomposition process) from LES of flow over figure 1(c) homogeneous roughness. (a) Shows low- and high-pass-filtered velocity, $\tilde{u} _{S}^{\prime }(y_{i},t)$ (black line) and $\tilde{u} _{L}^{\prime }(y_{i},t)$ (orange line), respectively, at inner location; (b) shows low-pass-filtered envelope of small-scale inner velocity, $E_{L}(\tilde{u} _{S}^{\prime }(y_{i},t))$ (black line), and time-shifted low-pass-filtered outer velocity, $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y_{i}))$ (orange line). Grey shading is added for illustration of AM effects during passage of a coherent high-momentum region. For this figure, the inner and outer elevations are $y_{i}/{\it\delta}=0.004$ and $y_{O}/{\it\delta}=0.1$ , respectively.

To advance the discussion, we now present results of the AM decoupling procedure, as outlined in Mathis et al. (Reference Mathis, Hutchins and Marusic2009a ) and briefly repeated in § 1.1. We have applied the decoupling procedure to time series for the HR and CR cases at elevations that capture the roughness sublayer ( $y_{i}/y_{RSL}<1$ ) and inertial layer ( $y_{O}/y_{RSL}>1$ ). For case HR, figure 3 shows time series of AM parameters at $y_{i}/{\it\delta}\approx 0.004$ , based on the large-scale modulating signal at $y_{O}/{\it\delta}\approx 0.1$ (recall table 1, which shows that $y_{i}/{\it\delta}<y_{RSL}/{\it\delta}<y_{O}/{\it\delta}$ ). Figure 3(a) shows the small- and large-scale signals, $\tilde{u} _{S}^{\prime }(y_{i},t)$ and $\tilde{u} _{L}^{\prime }(y_{i},t)$ . Added for illustration is grey shading where $\tilde{u} _{S}^{\prime }(y_{i},t)$ exhibits high variance. It is clear that periods when $\tilde{u} _{S}^{\prime }(y_{i},t)$ varies dramatically correspond with $\tilde{u} _{L}^{\prime }(y_{i},t)>0$ . In contrast, during the interval $30\lesssim tU_{0}/{\it\delta}\lesssim 100$ , $\tilde{u} _{L}^{\prime }(y_{i},t)<0$ and $\tilde{u} _{S}^{\prime }(y_{i},t)$ is relatively quiescent. Following Mathis et al. (Reference Mathis, Hutchins and Marusic2009a ), we show the low-pass-filtered envelope, $E_{L}(\tilde{u} _{S}^{\prime }(y_{i},t))$ , in figure 3(b). Here, $E_{L}(\tilde{u} _{S}^{\prime }(y_{i},t))$ is important, since it is absent in small-scale effects in the ‘carrier’ and illustrates only the influence of larger-scale inertial layer flow structures. To objectively demonstrate this, figure 3(b) shows $E_{L}(\tilde{u} _{S}^{\prime }(y_{i},t))$ and $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y_{i}))$ , where the ${\it\delta}{\it\tau}(y_{O};y_{i})$ correction accounts for the advective lag evident in figure 2. Recently, Anderson et al. (Reference Anderson, Li and Bou-Zeid2015) showed that a shear-normalized advective lag ( $u_{{\it\tau}}{\it\delta}^{-1}$ ; Stevens et al. Reference Stevens, Wilczek and Meneveau2014) can be determined a priori using canonical attributes of coherent structures in the flow. If an advective velocity, $U_{0}$ , and streamwise correlation length, $L_{s}$ , are known, these can be combined to determine the time before a uniform momentum parcel of fluid will advect past a sampling point. The logarithmic law and inclination angle are thus introduced:

where ${\it\kappa}=0.4$ is the von Kármán constant ( ${\it\kappa}=0.4$ ), $l$ is the aerodynamic roughness length, and ${\it\gamma}$ is the inclination angle of coherent structures ( ${\it\gamma}=15^{\circ }$ ) (Coceal et al. Reference Coceal, Dobre, Thomas and Belcher2007; Hong et al. Reference Hong, Katz and Schultz2011, Reference Hong, Katz, Meneveau and Schultz2012). Dividing length by advective velocity and applying the shear normalization factor, $u_{{\it\tau}}{\it\delta}^{-1}$ , yields an advective lag:

Equation (3.2) predictions compared favourably against the actual advective lag, based on computation of the correlation between fluctuating streamwise velocity at different elevations. For all subsequent applications of the (3.2) time lag, we remove the $u_{{\it\tau}}{\it\delta}^{-1}$ normalization factor; note also that time, $t$ , is normalized by $u_{{\it\tau}}{\it\delta}^{-1}$ when combined with ${\it\delta}{\it\tau}$ . Figure 3 illustrates that the passage of LMRs ( $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y_{i}))<0$ ) and HMRs ( $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y_{i}))>0$ ) in the inertial layer corresponds closely with the inner-layer modulation of small scales as evidenced by $E_{L}(\tilde{u} _{S}(y_{i},t))<0$ and $E_{L}(\tilde{u} _{S}(y_{i},t))>0$ , respectively.

Figure 4.

Time series of velocity and AM parameters (recall § 1.1 decomposition process) from LES of flow over figure 1(a) cubic topography, at position $\{x_{4},z_{4}\}$ in figure 1(b). (a,c) Show low- and high-pass-filtered velocity, $\tilde{u} _{S}^{\prime }(y_{i},t)$ (black line) and $\tilde{u} _{L}^{\prime }(y_{i},t)$ (orange line), respectively, at inner location; (b,d) show low-pass-filtered envelope of small-scale inner velocity, $E_{L}(\tilde{u} _{S}^{\prime }(y_{i},t))$ (black line), and time-shifted low-pass-filtered outer velocity, $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y_{i}))$ (orange line). In (a,b), grey shading is added for illustration of AM effects during passage of a coherent high-momentum region. For this figure, the inner and outer elevations are $y_{i}/{\it\delta}=0.254$ and $y_{O}/{\it\delta}=0.5$ , respectively.

For flow over case CR, figure 4 shows results of the AM decoupling procedure at position $\{x_{4},z_{4}\}$ (see figure 1 for a schematic). The results are entirely consistent with those for case HR (figure 3). During the passage of a HMR (i.e. figure 4 a,b: $5\lesssim tU_{0}/{\it\delta}\lesssim 40$ and $50\lesssim tU_{0}/{\it\delta}\lesssim 125$ ), the envelope of the small-scale inner velocity illustrates a clear modulation by the inertial layer velocity. Likewise, during quiescent periods when LMRs pass through the domain (figure 4 c,d), there is also convincing evidence that $E_{L}(\tilde{u} _{S}^{\prime }(y_{i},t))$ is modulated by $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y_{i}))>0$ . For brevity, similarly formatted figures for positions $x_{2}$ and $x_{3}$ have been excluded here, although they portray the same relations. We emphasize that $\tilde{u} _{L}^{\prime }(y_{O},t)$ would be well resolved – based on decomposition of $x{-}y$ Reynolds shear stresses, $\langle u^{\prime }v^{\prime }\rangle =\langle \tilde{u} ^{\prime }\tilde{v}^{\prime }\rangle +\langle {\it\tau}_{xy}\rangle$ – for $y_{O}/{\it\delta}\gtrsim 0.1$ during LES (see, for example, figure 4 of Porté-Agel et al. (Reference Porté-Agel, Meneveau and Parlange2000), figure 4 of Bou-Zeid et al. (Reference Bou-Zeid, Meneveau and Parlange2005), or figure 14 of Anderson & Meneveau (Reference Anderson and Meneveau2011)).

Figure 5.

Statistics for flow over case HR. (a) Shows premultiplied spectral density of streamwise velocity, $k_{x}E_{\tilde{u} ^{\prime }\tilde{u} ^{\prime }}/u_{{\it\tau}}^{2}$ , for LES with $N=128$ . (b,c) Show vertical profiles of single- and two-point AM correlation coefficients, $R(y;y)$ (thin solid lines, (3.3)) and $R(y;y_{O})$ (thick solid lines, (3.4)), respectively, for flow over case HR. In (a,b), wall-normal elevation is scaled with channel depth, ${\it\delta}$ . In (c), wall-normal elevation is scaled with LES filter width, ${\it\Delta}$ . Blue and black lines correspond with $N=96$ and $N=128$ , respectively. In (a,b), solid vertical orange line corresponds to the (approximate) elevation of the roughness element height, $y_{0}/h=0.1$ (Garratt Reference Garratt1994) (see also (2.4)), dashed vertical orange line denotes $2h/{\it\delta}\approx y_{RSL}/{\it\delta}$ (dictated by (2.4)), and vertical solid black line denotes reference elevation, $y_{O}/{\it\delta}$ (see table 1). In (a), solid horizontal blue lines at scales of interest are shown (with annotations), and the horizontal orange line at ${\it\lambda}_{x}/{\it\delta}=1$ is shown since it is the filter scale in the present AM decomposition (see § 1.1). In (b), black squares denote smooth-wall channel flow $R(y;y)$ , taken from (Mathis et al. Reference Mathis, Monty, Hutchins and Marusic2009b ) (their figure 3a), and grey shading is added to highlight the logarithmic region.

Figure 5(a) shows a contour of premultiplied spectral density of resolved (LES) streamwise fluctuating velocity, $k_{x}E_{\tilde{u} ^{\prime }\tilde{u} ^{\prime }}/u_{{\it\tau}}^{2}$ , where $k_{x}=2{\rm\pi}/{\it\lambda}_{x}$ is wavenumber and ${\it\lambda}_{x}$ is wavelength (Hutchins & Marusic Reference Hutchins and Marusic2007b ). Here, ${\it\lambda}_{x}$ is recovered from the time series with Taylor’s hypothesis via selection of a convection velocity, $U_{c}$ . In this study, we selected $U_{c}=U_{0}$ . Mathis et al. (Reference Mathis, Hutchins and Marusic2009a ) considered $U_{c}$ effects on premultiplied spectral density contours and correlation between $\tilde{u} _{L}^{\prime }(y,t)$ and $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ , $R(y;y)$ (discussion to follow). They reported that changing $U_{c}$ ‘shifts’ the distribution of $k_{x}E_{\tilde{u} ^{\prime }\tilde{u} ^{\prime }}/u_{{\it\tau}}^{2}$ across ${\it\lambda}_{x}$ , but that a robust scale separation remains and, importantly, that ${\it\lambda}_{x}/{\it\delta}=1$ is clearly between the wavelengths corresponding with the inner and outer peaks in energy content. They also demonstrated that $U_{c}$ has a relatively modest impact on the resulting $R(y;y)$ profile. The figure 5(a) contour demonstrates a clear outer peak across $2\lesssim {\it\lambda}_{x}/{\it\delta}\lesssim 5$ and $y/{\it\delta}\approx 0.16$ , which is consistent with the signature of large-scale motions (LSMs, Hutchins & Marusic Reference Hutchins and Marusic2007a ; Mathis et al. Reference Mathis, Hutchins and Marusic2009a ; Fang & Porté-Agel Reference Fang and Porté-Agel2015). Note, however, that the present computational domain does not capture content associated with very-large-scale motions (VLSMs, or so-called ‘superstructures’, Hutchins & Marusic Reference Hutchins and Marusic2007a ; Fang & Porté-Agel Reference Fang and Porté-Agel2015). Thus, we do not resolve the plateau and ‘second peak’ at ${\it\lambda}_{x}/{\it\delta}\approx 20$ due to VLSMs which occur at these asymptotic Reynolds number conditions (Fang & Porté-Agel Reference Fang and Porté-Agel2015). In this work, we argue that LSMs – which are resolved in the domain – represent the ‘building blocks’ of VLSMs (in accordance with the literature from Kim & Adrian Reference Kim and Adrian1999; Balakumar & Adrian Reference Balakumar and Adrian2007; Dennis & Nickels Reference Dennis and Nickels2011a ,Reference Dennis and Nickels b ; Ahn et al. Reference Ahn, Lee, Lee, Kang and Sung2015; Hellström, Ganapathisubramania & Smits Reference Hellström, Ganapathisubramania and Smits2015). Thus, since LSMs are resolved, they embody the physics imparted by VLSMs since the latter form via quasistreamwise coalescence of the former. With this, we select ${\it\lambda}_{x}/{\it\delta}=1$ as a separating scale for the present rough-wall LES, since scales exceeding ${\it\lambda}_{x}/{\it\delta}$ will be related to LSMs and application of the AM decoupling procedure will therefore reveal their effect on sub- ${\it\delta}$ motions. For illustration, we added the LES grid-filter width, ${\it\Delta}/{\it\delta}$ , to figure 5(a) ( ${\it\Delta}=({\it\Delta}_{x}{\it\Delta}_{y}{\it\Delta}_{z})^{1/3}$ , Deardorff Reference Deardorff1974; Scotti, Meneveau & Lilly Reference Scotti, Meneveau and Lilly1993; Bou-Zeid et al. Reference Bou-Zeid, Meneveau and Parlange2005), which demonstrates the spatial scales resolved on the LES mesh. The figure 5(a) spectral density provides a guide for selecting the reference elevation, which we chose here as $y_{O}/{\it\delta}=0.16$ . Thus, we can identify AM of $\tilde{u} _{S}^{\prime }(y,t)$ by $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{O};y))$ , where $y_{O}$ captures large-scale energy-containing structures (please see continuing discussion below).

To determine the ‘strength’ of the AM effect, we follow the Mathis et al. (Reference Mathis, Hutchins and Marusic2009a ) approach by determining the correlation between $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ and the large-scale signal. We can compute correlations based on single-point:

or two-point measurements:

Thus, (3.4) can be used to determine AM of $\tilde{u} _{S}^{\prime }(y_{i},t)$ by $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y_{i};y_{O}))$ . Please see (3.2) and accompanying discussion for elaboration about the advective correction, ${\it\delta}{\it\tau}(y;y_{O})$ , which is essential to account for the time delay between passage of inclined coherent structures in the outer layer and their ‘footprint’ in the roughness sublayer (Piomelli et al. Reference Piomelli, Ferziger, Moin and Kim1989; Marusic et al. Reference Marusic, Kunkel and Porté-Agel2001; Coceal et al. Reference Coceal, Dobre, Thomas and Belcher2007; Marusic & Heuer Reference Marusic and Heuer2007; Mathis et al. Reference Mathis, Hutchins and Marusic2009a ; Wu & Christensen Reference Wu and Christensen2010; Anderson et al. Reference Anderson, Li and Bou-Zeid2015). Table 1 summarizes the reference elevations with respect to channel depth and roughness element scale; in all cases, $y_{O}$ has been carefully selected to ensure that correlations are evaluated with respect to inertial layer flow processes (i.e.  $y_{O}/h>2$ or $y_{O}/y_{RSL}>1$ ). And, although $2\lesssim y_{RSL}/h\lesssim 5$ (Grass Reference Grass1971; Raupach et al. Reference Raupach, Antonia and Rajagopalan1991; Mejia-Alvarez & Christensen Reference Mejia-Alvarez and Christensen2010) is the widely accepted range for $y_{RSL}$ , we note also that for the present CR case (Cheng & Castro Reference Cheng and Castro2002), local mean streamwise velocity profiles all converge to a plane-averaged value for $y/h\lesssim 2$ (Coceal et al. Reference Coceal, Dobre, Thomas and Belcher2007; Anderson et al. Reference Anderson, Li and Bou-Zeid2015).

Figure 5(b,c) shows profiles of $R(y;y)$ and $R(y;y_{O})$ for case HR, wherein wall-normal elevation is scaled against channel depth, ${\it\delta}$ (a), and LES filter width, ${\it\Delta}$ (b), respectively. In (a), the $R(y;y)$ and $R(y;y_{O})$ profiles are similar for $y/{\it\delta}\lesssim 0.2$ , above which the profiles diverge, with $R(y;y_{O})$ declining to zero. This divergence shows that $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y;y_{O}))$ is more weakly correlated with $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ for increasing $y$ (while, at the same elevation, $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ and $\tilde{u} _{L}^{\prime }(y,t)$ remain quite strongly correlated). For $y/{\it\delta}\lesssim 0.05$ – where the range of turbulent scales owing to shear production of turbulence is greatest and subgrid-scale stresses are maximum – both correlations are elevated. Moreover, close inspection illustrates that $R(y;y)>R(y;y_{O})$ for $y/{\it\delta}\lesssim 0.05$ , which is a physically meaningful outcome of correlating $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ against $\tilde{u} _{L}^{\prime }(y,t)$ , instead of $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y;y_{O}))$ . Nonetheless, comparison of the rough-wall $R(y;y)$ profile against that for a smooth-wall channel (black squares, taken from Mathis et al. (Reference Mathis, Monty, Hutchins and Marusic2009b ), their figure 3a) illustrates that the correlation is stronger in the presence of roughness. Note that ${\it\delta}/h\sim O(10^{2})$ for the present HR case (assuming that $h/l\approx 10$ , as per (2.4)), thus rendering this case a ‘roughness’ (Jimenez Reference Jimenez2004; Flack & Schultz Reference Flack and Schultz2010; Hong et al. Reference Hong, Katz, Meneveau and Schultz2012) with flow statistics that will conform with Townsend’s wall similarity hypothesis (Townsend Reference Townsend1976; Wu & Christensen Reference Wu and Christensen2007, Reference Wu and Christensen2010). One of course expects this since the roughness sublayer is parameterized via the equilibrium logarithmic law through the roughness length, $l$ ((2.2) and (2.3)). It is therefore interesting that $R(y;y)$ and $R(y;y_{O})$ in the roughness sublayer and outer (inertial) layer exceed the smooth-wall values.

Figure 6.

Premultiplied spectral density of streamwise velocity, $k_{x}E_{\tilde{u} ^{\prime }\tilde{u} ^{\prime }}/u_{{\it\tau}}^{2}$ , for flow over case CR at positions indicated in figure insets. Spectral density shown here for LES with $N=128$ . Wall-normal elevation is scaled with channel depth, ${\it\delta}$ . Solid vertical orange line corresponds to the roughness element height, $h/{\it\delta}=0.25$ , dashed vertical orange line denotes $2h/{\it\delta}$ , and vertical solid black line denotes reference elevation, $y_{O}$ . For reference, solid horizontal blue lines at scales of interest are shown (with appropriate annotations), and the horizontal orange line at ${\it\lambda}_{x}/{\it\delta}=1$ is shown since it is the filter scale in the present AM decomposition (see § 1.1).

Figure 6 shows $k_{x}E_{\tilde{u} ^{\prime }\tilde{u} ^{\prime }}/u_{{\it\tau}}^{2}$ contours for flow over case CR (figure insets illustrate spatial position). We again have included horizontal lines at reference scales ( ${\it\Delta}/{\it\delta}$ , $h/{\it\delta}$ , and $L_{x}/{\it\delta}$ ), which provide perspective on energy content at different scales as it relates to wall-normal elevation. At $\{x_{1},z_{1}\}$ , there is no data for $y/{\it\delta}\leqslant h/{\it\delta}$ (owing to the ‘immersed points’). Immediately above this, however, for $5\times 10^{-1}\lesssim {\it\lambda}_{x}/{\it\delta}\lesssim 1.2\times 10^{0}$ , $k_{x}E_{\tilde{u} ^{\prime }\tilde{u} ^{\prime }}/u_{{\it\tau}}^{2}$ exhibits a peak that is associated with production of turbulence from the roughness element. At downwind positions, this peak remains but occurs at lower elevations, which is due to the shearing layer present at the approximate cube height. At $\{x_{1},z_{1}\}$ , there is also a second peak located at ${\it\lambda}_{x}/{\it\delta}\approx 1.8\times 10^{0}$ , for $0.3\leqslant y/{\it\delta}\leqslant 0.7$ , which indicates the presence of large structures (scale exceeding ${\it\delta}$ ) in the domain. In fact, energy content at this wavelength and elevation is present at all locations (see figure 6 b–d), and this has guided our selection of $y_{O}/{\it\delta}=0.6$ (i.e.  $y_{O}/y_{RSL}>1$ , and $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y;y_{O}))$ contains significant large-scale energy content). In DNS studies of flow over case CR, Coceal et al. (Reference Coceal, Dobre, Thomas and Belcher2007) have shown that vertical profiles of time-averaged velocity at positions $\{x_{1},x_{1}\}$ , $\{x_{2},x_{2}\}$ , $\{x_{3},x_{3}\}$ , and $\{x_{4},x_{4}\}$ (see figure 1 b) all merge with the plane- and time-averaged vertical profile slightly above the cubes (their figure 5a); in LES modelling of flow over the same arrangement, we (Anderson et al. Reference Anderson, Li and Bou-Zeid2015) also observed this trend. Thus, $y_{O}/{\it\delta}=0.6$ is in the inertial (logarithmic) layer, and therefore the large scales (retrieved from low-pass filtering) are low- and high-momentum regions and it is their influence on the roughness sublayer that is of interest here. Figure 6 also contains important information on downwind evolution of turbulence below the cube heights. At $\{x_{2},z_{2}\}$ , there is limited content at smaller scales for $y/{\it\delta}\lesssim 0.2$ . At $\{x_{3},z_{3}\}$ , an inner peak has emerged at ${\it\lambda}_{x}/{\it\delta}\approx 8\times 10^{-1}$ for $10^{-2}\lesssim y/{\it\delta}\lesssim 1.5\times 10^{-1}$ , which appears to broaden (and lower) at $\{x_{4},z_{4}\}$ . As discussed below, sub- ${\it\delta}$ spectral density below the cube height has important implications for correlating $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y/h<1;y_{O}))$ with $\tilde{u} _{S}^{\prime }(y/h<1,t)$ .

Figure 7.

Vertical profiles of single- and two-point AM correlation coefficients, $R(y;y)$ (thin solid lines, (3.3)) and $R(y;y_{O})$ (thick solid lines, (3.4)), respectively, for flow over case CR. Wall-normal elevation is scaled with channel depth, ${\it\delta}$ . Grey, blue, and black lines correspond with $N=64$ , $N=96$ , and $N=128$ , respectively. Solid vertical orange line corresponds to the roughness element height, $h/{\it\delta}=0.25$ , dashed vertical orange line denotes $2h/{\it\delta}$ , and vertical solid black line denotes reference elevation, $y_{O}$ .

Figure 8.

Vertical profiles of single- and two-point AM correlation coefficients, $R(y;y)$ (thin solid lines, (3.3)) and $R(y;y_{O})$ (thick solid lines, (3.4)), respectively. Here, wall-normal elevation is scaled with LES filter width, ${\it\Delta}$ , which provides a means to compare across LES cases and to provide more detail on the correlation profiles in close proximity to the cube height. Grey, blue, and black lines correspond with $N=64$ , $N=96$ , and $N=128$ , respectively.

Figures 7 and 8 show $R(y;y)$ and $R(y;y_{O})$ profiles for the CR case at spatial positions $\{x_{1},z_{1}\}$ , $\{x_{2},z_{2}\}$ , $\{x_{3},z_{3}\}$ , and $\{x_{4},z_{4}\}$ (see figure insets). Figure 7 shows the correlations against $y/{\it\delta}$ (with $h/{\it\delta}$ , $y_{RSL}/{\it\delta}$ , and $y_{O}/{\it\delta}$ included for reference). Since these results are from LES with an IBM that resolves the roughness sublayer, they reveal the extent to which local flow processes within the sublayer influence the correlations. Consider, for example, the profiles at positions $\{x_{2},z_{2}\}$ and $\{x_{3},z_{3}\}$ , which demonstrate that $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ is not correlated with $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y;y_{O}))$ for $y/{\it\delta}\lesssim 0.1$ (or $y/h\lesssim 0.4$ ). At $\{x_{4},z_{4}\}$ , $R(y;y_{O})$ is substantially higher below the cubes. However, at all positions, $R(y;y_{O})\approx 0.4$ at the cube height, $y/h=1$ . It is at this height that the $x{-}y$ turbulent (Reynolds) shear stresses are highest (Ghisalberti Reference Ghisalberti2009), which is important in terms of modelling surface drag. In contrast, the single-point ( $R(y;y)$ ) correlations are elevated at positions $\{x_{2},z_{2}\}$ and $\{x_{3},z_{3}\}$ , and negative at $\{x_{4},z_{4}\}$ . At $\{x_{2},z_{2}\}$ and $\{x_{3},z_{3}\}$ , this confirms that the small-scale content of roughness sublayer secondary flows (due to intermittent vortex shedding from cubes, recirculating flow, and downwind migration of Kelvin–Helmholtz vortices associated with ‘Stuart’ vortices) is positively correlated with its large-scale content (though not adequately strong to be correlated with large-scale content in the inertial layer). In contrast, $R(y;y)<0$ at $\{x_{4},z_{4}\}$ for $0\leqslant y/{\it\delta}\lesssim 0.05$ , which is due to mean recirculating flow upwind of the next cube. We note, however, that above the cubes (but still in the roughness sublayer, i.e.  $0.25\lesssim y/{\it\delta}\lesssim 0.5$ ) the correlation profiles are all relatively similar (based on single- and two-point correlation). We emphasize fundamental differences between case CR (channel flow with topography) and smooth-wall turbulent boundary layers, which were first used to demonstrate the presence of AM in wall turbulence (Marusic et al. Reference Marusic, Mathis and Hutchins2010). Indeed, turbulence statistics for case CR will not exhibit outer-layer similarity, and thus coherent structures present for case CR may not have the spatial footprint of those present for case HR or smooth walls. Nonetheless, through application of the AM decoupling procedure (Mathis et al. Reference Mathis, Hutchins and Marusic2009a ), we demonstrate clear correlation between inertial and roughness sublayer processes.

In order to gain more insight on the roughness sublayer, we have also plotted the CR case correlation profiles against $(y-h)/{\it\Delta}$ , where ${\it\Delta}=({\it\Delta}_{x}{\it\Delta}_{y}{\it\Delta}_{z})^{1/3}$ is the LES grid resolution (Deardorff Reference Deardorff1974; Scotti et al. Reference Scotti, Meneveau and Lilly1993; Bou-Zeid et al. Reference Bou-Zeid, Meneveau and Parlange2005) and $h$ represents the elevation at which the streamwise–wall-normal Reynolds (total) shearing stresses are maximum (Coceal et al. Reference Coceal, Dobre, Thomas and Belcher2007) (although, in reality, the elevation corresponding with maximum $\widetilde{u^{\prime }w^{\prime }}/u_{{\it\tau}}^{2}$ exhibits spatial variation in these canopy flows, Anderson & Chamecki Reference Anderson and Chamecki2014). Normalizing by ${\it\Delta}$ is helpful in LES studies that feature resolution testing (Bou-Zeid et al. Reference Bou-Zeid, Meneveau and Parlange2005). It is clear firstly that the LES results do not exhibit resolution dependence. Moreover, this plotting convention demonstrates generally positive correlation between $E_{L}(\tilde{u} _{S}^{\prime }(y,t))$ and $\tilde{u} _{L}^{\prime }(y_{O},t+{\it\delta}{\it\tau}(y;y_{Ref.}))$ , where the maximum values occur slightly above the cube height for positions $\{x_{1},z_{1}\}$ , $\{x_{2},z_{2}\}$ , and $\{x_{3},z_{3}\}$ . The large- and small-scale contents exhibit strong correlation at the same elevations ( $R(y;y)$ ), with maximum values also occurring at the cube height, $(y-h)/{\it\Delta}=0$ .